Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

Kei Kondo

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

Differential Geometry 2021-06-30 v9 Functional Analysis Geometric Topology Optimization and Control

Authors: Kei Kondo

Abstract

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$ , where $\dim M \geq \dim N$ , has no singular points on $M$ in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

Keywords

lipschitz maps ehresmann fibrations smooth approximation reeb sphere theorem singular points compact riemannian manifolds

Cite

@article{arxiv.1811.04340,
  title  = {Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions},
  author = {Kei Kondo},
  journal= {arXiv preprint arXiv:1811.04340},
  year   = {2021}
}

Comments

Minor corrections. Theorem 1.7 has been improved since the version 11 (i.e., an assumption of the theorem has been removed). 28 pages, no figures

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

Abstract

Keywords

Cite

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